Hi. I agree with pretty much everything you said. The argument on its own has holes. I can give you an answer that stands outside of the argument though.

Each individual Christian claim either falls under the set of topics that (1) can be found with natural reason alone (which includes science), or (2) needs to be supported by divine revelation such as the Bible.

If (1), then the argument stands. If (2), then fortunately arguments can be made to defend that the Bible is a reliable source: If all the verifiable claims from a source or method are verified to be true, then, by induction, it is reasonable to conclude that the remaining unverifiable claims from that same source or method are also true.

As an analogy: If all the planets we have observed so far are round, then it is reasonable to predict that the next planet we discover will also be round.